SOLUTION: Find all real or imaginary solutions to the equation. x-1/x+2=2x-3/x+4

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Question 78015: Find all real or imaginary solutions to the equation.
x-1/x+2=2x-3/x+4
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-1%29%2F%28x%2B2%29=%282x-3%29%2F%28x%2B4%29

%28%28x%2B2%29%28x%2B4%29%29%28%28x-1%29%2F%28x%2B2%29%29=%28%282x-3%29%2F%28x%2B4%29%29%28%28x%2B2%29%28x%2B4%29%29Multiply both sides by the LCD %28x%2B2%29%28x%2B4%29
%28cross%28%28x%2B2%29%29%28x%2B4%29%29%28%28x-1%29%2Fcross%28%28x%2B2%29%29%29=%28%282x-3%29%2Fcross%28%28x%2B4%29%29%29%28%28x%2B2%29cross%28%28x%2B4%29%29%29 Cancel the common terms
So we end up with this:
%28x%2B4%29%28x-1%29=%282x-3%29%28x%2B2%29

x%5E2%2B3x-4=2x%5E2%2Bx-6 foil the parenthesis

0=x%5E2-2x-2 Get all terms to one side
Now use the quadratic formula to solve for x:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-2x%2B-2+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A1%2A-2=12.

Discriminant d=12 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+12+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+12+%29%29%2F2%5C1+=+2.73205080756888
x%5B2%5D+=+%28-%28-2%29-sqrt%28+12+%29%29%2F2%5C1+=+-0.732050807568877

Quadratic expression 1x%5E2%2B-2x%2B-2 can be factored:
1x%5E2%2B-2x%2B-2+=+1%28x-2.73205080756888%29%2A%28x--0.732050807568877%29
Again, the answer is: 2.73205080756888, -0.732050807568877. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-2+%29


So we have 2 real solutions:
x=-0.732050807568877 or x=2.73205080756888